Questo documento va caricato in Moodle come cognome-nome-esX.Rmd dove X è il numero dell’esercizio (1, 2, 3 o 4).

1 Soluzione quesito 1

scrivere qui la soluzione del primo quesito

2 Soluzione quesito 2

scrivere qui la soluzione del secondo quesito

3 Soluzione quesito 3

scrivere qui la soluzione del terzo quesito (sempre che ve ne siano tre, altrimenti cancellare questa sezione)

4 Prototipazione Regressione

5 Inserimento dei dati e Analizziamo

    dataset <- read.csv('efficienza.csv')
    # x_name <- processi 
    # y_name <- dimensione
    plot(processi ~ dimensione, data = dataset)

    summary(dataset)
##     processi        dimensione  
##  Min.   :  1.00   Min.   : 6.0  
##  1st Qu.: 12.00   1st Qu.: 8.0  
##  Median : 26.00   Median :14.0  
##  Mean   : 35.52   Mean   :13.1  
##  3rd Qu.: 48.00   3rd Qu.:17.0  
##  Max.   :100.00   Max.   :20.0
    outliers_range <- 80
    outliers <- which(dataset$processi > outliers_range) # processi è la x_name
    summary(outliers)
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##    8.00    8.75   13.50   16.00   20.75   29.00

6 Stimiamo il modello

6.1 Stima della retta di regressione:

  • Normale
    mod <- lm(processi ~ dimensione, data = dataset) 
    plot(processi ~ dimensione , data=dataset)
    curve(coef(mod)[1] + coef(mod)[2] * x + coef(mod)[3] * x ^ 2, from = min(dataset$processi), to = max(dataset$processi), add = TRUE, col = "blue")
    abline(mod, col = 'red')

    summary(mod)
## 
## Call:
## lm(formula = processi ~ dimensione, data = dataset)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -26.392 -19.051 -11.370   4.014  69.289 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)   
## (Intercept)   57.261     16.551   3.460  0.00181 **
## dimensione    -1.659      1.199  -1.384  0.17783   
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 27.94 on 27 degrees of freedom
## Multiple R-squared:  0.0662, Adjusted R-squared:  0.03162 
## F-statistic: 1.914 on 1 and 27 DF,  p-value: 0.1778
  • Analisi con i residui:
    par(mfrow = c(1,2))
    plot(residuals(mod) ~ dimensione, data=dataset)
    abline(h=0, col = "red")
    qqPlot(residuals(mod))

## [1] 8 9
    par(mfrow = c(1,1))
  • Analisi senza i residui:
    mod2 <- lm(processi ~ dimensione , data = dataset, subset=-outliers) 
    plot(processi ~ dimensione , data=dataset, subset= -outliers)

    summary(mod2)
## 
## Call:
## lm(formula = processi ~ dimensione, data = dataset, subset = -outliers)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -11.9634  -4.6211   0.5299   4.0233  13.1877 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  74.6613     4.2659   17.50 8.67e-15 ***
## dimensione   -3.8356     0.3217  -11.92 2.51e-11 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 7.034 on 23 degrees of freedom
## Multiple R-squared:  0.8608, Adjusted R-squared:  0.8547 
## F-statistic: 142.2 on 1 and 23 DF,  p-value: 2.508e-11
    plot(processi ~ dimensione, data = dataset)
    curve(coef(mod)[1] + coef(mod)[2] * x + coef(mod)[3] * x ^ 2, from = min(dataset$processi), to = max(dataset$processi), add = TRUE, col = "blue")
    curve(coef(mod2)[1] + coef(mod2)[2] * x + coef(mod2)[3] * x ^ 2, from = min(dataset$processi), to = max(dataset$processi), add = TRUE, col = "green")
    abline(mod, col = 'red')

    par(mfrow = c(1,2))
    plot(residuals(mod2) ~ dimensione[-outliers], data=dataset)
    abline(h=0, col = "red")
    qqPlot(residuals(mod2))

## [1] 3 1
    par(mfrow = c(1,1))

6.2 Stima Quadratica:

    mod_quad <- lm(processi ~ dimensione + I(dimensione ^ 2), data = dataset) 
    plot(processi ~ dimensione + I(dimensione ^ 2), data=dataset)

    curve(coef(mod_quad)[1] + coef(mod_quad)[2] * x + coef(mod_quad)[3] * x ^ 2, from = min(dataset$processi), to = max(dataset$processi), add = TRUE, col = "blue")

    summary(mod_quad)
## 
## Call:
## lm(formula = processi ~ dimensione + I(dimensione^2), data = dataset)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -27.930 -19.101  -9.357   3.668  70.423 
## 
## Coefficients:
##                 Estimate Std. Error t value Pr(>|t|)
## (Intercept)      78.5531    54.4679   1.442    0.161
## dimensione       -5.4942     9.4102  -0.584    0.564
## I(dimensione^2)   0.1521     0.3700   0.411    0.684
## 
## Residual standard error: 28.38 on 26 degrees of freedom
## Multiple R-squared:  0.07223,    Adjusted R-squared:  0.0008619 
## F-statistic: 1.012 on 2 and 26 DF,  p-value: 0.3773
  • Analisi con i residui:
    par(mfrow = c(1,2))
    plot(residuals(mod_quad) ~ dimensione, data=dataset)
    abline(h=0, col = "red")
    qqPlot(residuals(mod_quad))

## [1] 8 9
    par(mfrow = c(1,1))
  • Analisi senza i residui:
    mod_quad_2 <- lm(processi ~ dimensione + I(dimensione ^ 2), data = dataset, subset=-outliers) 
    plot(processi ~ dimensione + I(dimensione ^ 2), data=dataset, subset= -outliers)

    summary(mod_quad_2)
## 
## Call:
## lm(formula = processi ~ dimensione + I(dimensione^2), data = dataset, 
##     subset = -outliers)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -14.8440  -3.4158   0.5015   4.0487  10.5842 
## 
## Coefficients:
##                  Estimate Std. Error t value Pr(>|t|)    
## (Intercept)     104.33360   12.36306   8.439  2.4e-08 ***
## dimensione       -9.18934    2.13998  -4.294 0.000294 ***
## I(dimensione^2)   0.21240    0.08412   2.525 0.019283 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 6.333 on 22 degrees of freedom
## Multiple R-squared:  0.892,  Adjusted R-squared:  0.8822 
## F-statistic:  90.9 on 2 and 22 DF,  p-value: 2.321e-11
    plot(processi ~ dimensione, data = dataset)
    curve(coef(mod_quad)[1] + coef(mod_quad)[2] * x + coef(mod_quad)[3] * x ^ 2, from = min(dataset$processi), to = max(dataset$processi), add = TRUE, col = "blue")
    curve(coef(mod_quad_2)[1] + coef(mod_quad_2)[2] * x + coef(mod_quad_2)[3] * x ^ 2, from = min(dataset$processi), to = max(dataset$processi), add = TRUE, col = "green")

    par(mfrow = c(1,2))
    plot(residuals(mod_quad_2) ~ dimensione[-outliers], data=dataset)
    abline(h=0, col = "red")
    qqPlot(residuals(mod_quad_2))

## 23  3 
## 20  3
    par(mfrow = c(1,1))

6.3 Stima Cubica:

    mod_cubica <- lm(processi ~ dimensione + I(dimensione ^ 3), data = dataset) 
    plot(processi ~ dimensione + I(dimensione ^ 2), data=dataset)

    curve(coef(mod_cubica)[1] + coef(mod_cubica)[2] * x + coef(mod_cubica)[3] * x ^ 2, from = min(dataset$processi), to = max(dataset$processi), add = TRUE, col = "blue")

    summary(mod_cubica)
## 
## Call:
## lm(formula = processi ~ dimensione + I(dimensione^3), data = dataset)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -27.450 -18.973  -9.846   3.653  70.223 
## 
## Coefficients:
##                  Estimate Std. Error t value Pr(>|t|)  
## (Intercept)     67.385126  38.510468   1.750    0.092 .
## dimensione      -3.056178   4.931776  -0.620    0.541  
## I(dimensione^3)  0.002757   0.009430   0.292    0.772  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 28.43 on 26 degrees of freedom
## Multiple R-squared:  0.06926,    Adjusted R-squared:  -0.002335 
## F-statistic: 0.9674 on 2 and 26 DF,  p-value: 0.3933
  • Analisi con i residui:
    par(mfrow = c(1,2))
    plot(residuals(mod_cubica) ~ dimensione, data=dataset)
    abline(h=0, col = "red")
    qqPlot(residuals(mod_cubica))

## [1] 8 9
    par(mfrow = c(1,1))
  • Analisi senza i residui:
    mod_cubica_2 <- lm(processi ~ dimensione + I(dimensione ^ 3), data = dataset, subset=-outliers) 
    plot(processi ~ dimensione + I(dimensione ^ 3), data=dataset, subset= -outliers)

    summary(mod_cubica_2)
## 
## Call:
## lm(formula = processi ~ dimensione + I(dimensione^3), data = dataset, 
##     subset = -outliers)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -14.3676  -3.3959   0.7948   3.9177  10.6041 
## 
## Coefficients:
##                  Estimate Std. Error t value Pr(>|t|)    
## (Intercept)     95.263293   8.670800  10.987 2.12e-10 ***
## dimensione      -6.684719   1.115278  -5.994 4.94e-06 ***
## I(dimensione^3)  0.005614   0.002124   2.643   0.0148 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 6.266 on 22 degrees of freedom
## Multiple R-squared:  0.8943, Adjusted R-squared:  0.8847 
## F-statistic: 93.09 on 2 and 22 DF,  p-value: 1.835e-11
    plot(processi ~ dimensione, data = dataset)
    curve(coef(mod_cubica)[1] + coef(mod_cubica)[2] * x + coef(mod_cubica)[3] * x ^ 2, from = min(dataset$processi), to = max(dataset$processi), add = TRUE, col = "blue")
    curve(coef(mod_cubica_2)[1] + coef(mod_cubica_2)[2] * x + coef(mod_cubica_2)[3] * x ^ 2, from = min(dataset$processi), to = max(dataset$processi), add = TRUE, col = "green")

    par(mfrow = c(1,2))
    plot(residuals(mod_cubica_2) ~ dimensione[-outliers], data=dataset)
    abline(h=0, col = "red")
    qqPlot(residuals(mod_cubica_2))

## 23  3 
## 20  3
    par(mfrow = c(1,1))

6.4 Stima logaritmica:

6.4.1 Per l’asse x :

  mod_logaritmica_x <- lm(log(processi) ~ dimensione, data = dataset) 
  plot(log(processi) ~ dimensione, data=dataset)
  abline(mod_logaritmica_x, col = "red")

  summary(mod_logaritmica_x)
## 
## Call:
## lm(formula = log(processi) ~ dimensione, data = dataset)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -2.57997 -0.39783 -0.05473  0.10326  1.91984 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  4.74800    0.54082   8.779 2.15e-09 ***
## dimensione  -0.12045    0.03919  -3.073   0.0048 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.913 on 27 degrees of freedom
## Multiple R-squared:  0.2591,   Adjusted R-squared:  0.2317 
## F-statistic: 9.445 on 1 and 27 DF,  p-value: 0.004798
  • Analisi con i residui
    par(mfrow = c(1,2))
    plot(residuals(mod_logaritmica_x) ~ dimensione, data=dataset)
    abline(h=0, col = "red")
    qqPlot(residuals(mod_logaritmica_x))

## [1] 6 9
  • Analisi senza i residui:
    mod_logaritmica_x2 <- lm(log(processi) ~ dimensione , data = dataset, subset=-outliers) 
    plot(log(processi) ~ dimensione , data=dataset, subset= -outliers)

    summary(mod_logaritmica_x2)
## 
## Call:
## lm(formula = log(processi) ~ dimensione, data = dataset, subset = -outliers)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -1.94733 -0.14401  0.08426  0.17129  0.64130 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  5.25862    0.30605  17.182 1.29e-14 ***
## dimensione  -0.18396    0.02308  -7.972 4.56e-08 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.5047 on 23 degrees of freedom
## Multiple R-squared:  0.7342, Adjusted R-squared:  0.7227 
## F-statistic: 63.54 on 1 and 23 DF,  p-value: 4.56e-08
    plot(log(processi) ~ dimensione, data = dataset)
    curve(coef(mod_logaritmica_x)[1] + coef(mod_logaritmica_x)[2] * x + coef(mod_logaritmica_x)[3] * x ^ 2, from = min(dataset$processi), to = max(dataset$processi), add = TRUE, col = "blue")
    curve(coef(mod_logaritmica_x2)[1] + coef(mod_logaritmica_x2)[2] * x + coef(mod_logaritmica_x2)[3] * x ^ 2, from = min(dataset$processi), to = max(dataset$processi), add = TRUE, col = "green")
    abline(mod_logaritmica_x, col = "red")

    par(mfrow = c(1,2))
    plot(residuals(mod_logaritmica_x2) ~ dimensione[-outliers], data=dataset)
    abline(h=0, col = "red")
    qqPlot(residuals(mod_logaritmica_x2))

##  6 16 
##  6 14
    par(mfrow = c(1,1))

6.4.2 Per l’asse y :

  mod_logaritmica_y <- lm(processi ~ log(dimensione), data = dataset) 
  plot(processi ~ log(dimensione), data=dataset)
  abline(mod_logaritmica_y, col = "red")

  summary(mod_logaritmica_y)
## 
## Call:
## lm(formula = processi ~ log(dimensione), data = dataset)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -26.790 -18.949 -10.488   3.764  69.821 
## 
## Coefficients:
##                 Estimate Std. Error t value Pr(>|t|)  
## (Intercept)        86.41      35.38   2.442   0.0214 *
## log(dimensione)   -20.28      13.95  -1.454   0.1575  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 27.84 on 27 degrees of freedom
## Multiple R-squared:  0.07261,  Adjusted R-squared:  0.03826 
## F-statistic: 2.114 on 1 and 27 DF,  p-value: 0.1575
  • Analisi con i residui
    par(mfrow = c(1,2))
    plot(residuals(mod_logaritmica_y) ~ dimensione, data=dataset)
    abline(h=0, col = "red")
    qqPlot(residuals(mod_logaritmica_y))

## [1] 8 9
  • Analisi senza i residui:
    mod_logaritmica_y2 <- lm(processi ~ log(dimensione) , data = dataset, subset=-outliers) 
    plot(processi ~ log(dimensione), data=dataset, subset= -outliers)

    summary(mod_logaritmica_y2)
## 
## Call:
## lm(formula = processi ~ log(dimensione), data = dataset, subset = -outliers)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -14.8395  -3.4371   0.2184   4.5486  11.1257 
## 
## Coefficients:
##                 Estimate Std. Error t value Pr(>|t|)    
## (Intercept)      137.800      8.527   16.16 4.76e-14 ***
## log(dimensione)  -45.185      3.426  -13.19 3.29e-12 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 6.443 on 23 degrees of freedom
## Multiple R-squared:  0.8832, Adjusted R-squared:  0.8781 
## F-statistic: 173.9 on 1 and 23 DF,  p-value: 3.289e-12
    plot(processi ~ log(dimensione), data = dataset)
    curve(coef(mod_logaritmica_y)[1] + coef(mod_logaritmica_y)[2] * x + coef(mod_logaritmica_y)[3] * x ^ 2, from = min(dataset$processi), to = max(dataset$processi), add = TRUE, col = "blue")
    curve(coef(mod_logaritmica_y2)[1] + coef(mod_logaritmica_y2)[2] * x + coef(mod_logaritmica_y2)[3] * x ^ 2, from = min(dataset$processi), to = max(dataset$processi), add = TRUE, col = "green")
    abline(mod_logaritmica_y, col = "red")

    par(mfrow = c(1,2))
    plot(residuals(mod_logaritmica_y2) ~ dimensione[-outliers], data=dataset)
    abline(h=0, col = "red")
    qqPlot(residuals(mod_logaritmica_y2))

## 23  3 
## 20  3
    par(mfrow = c(1,1))

6.4.3 Per entrambi gli assi:

  mod_logaritmica <- lm(log(processi) ~ log(dimensione), data = dataset) 
  plot(log(processi) ~ log(dimensione), data=dataset)
  abline(mod_logaritmica, col = "red")

  summary(mod_logaritmica)
## 
## Call:
## lm(formula = log(processi) ~ log(dimensione), data = dataset)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -2.6441 -0.4204 -0.0970  0.1637  1.8557 
## 
## Coefficients:
##                 Estimate Std. Error t value Pr(>|t|)    
## (Intercept)       6.6319     1.1675   5.680 4.93e-06 ***
## log(dimensione)  -1.3797     0.4603  -2.998  0.00578 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.9188 on 27 degrees of freedom
## Multiple R-squared:  0.2497,   Adjusted R-squared:  0.2219 
## F-statistic: 8.986 on 1 and 27 DF,  p-value: 0.00578
  • Analisi con i residui
    par(mfrow = c(1,2))
    plot(residuals(mod_logaritmica) ~ dimensione, data=dataset)
    abline(h=0, col = "red")
    qqPlot(residuals(mod_logaritmica))

## [1] 6 9
  • Analisi senza i residui:
    mod_logaritmica2 <- lm(log(processi) ~ log(dimensione) , data = dataset, subset=-outliers) 
    plot(log(processi) ~ log(dimensione), data=dataset, subset= -outliers)

    summary(mod_logaritmica2)
## 
## Call:
## lm(formula = log(processi) ~ log(dimensione), data = dataset, 
##     subset = -outliers)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -2.05396 -0.14892  0.07807  0.25270  0.60856 
## 
## Coefficients:
##                 Estimate Std. Error t value Pr(>|t|)    
## (Intercept)       8.1099     0.7047  11.508 5.07e-11 ***
## log(dimensione)  -2.0952     0.2832  -7.399 1.59e-07 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.5324 on 23 degrees of freedom
## Multiple R-squared:  0.7042, Adjusted R-squared:  0.6913 
## F-statistic: 54.75 on 1 and 23 DF,  p-value: 1.593e-07
    plot(log(processi) ~ log(dimensione), data = dataset)
    curve(coef(mod_logaritmica)[1] + coef(mod_logaritmica)[2] * x + coef(mod_logaritmica)[3] * x ^ 2, from = min(dataset$processi), to = max(dataset$processi), add = TRUE, col = "blue")
    curve(coef(mod_logaritmica2)[1] + coef(mod_logaritmica2)[2] * x + coef(mod_logaritmica2)[3] * x ^ 2, from = min(dataset$processi), to = max(dataset$processi), add = TRUE, col = "green")
    abline(mod_logaritmica, col = "red")

    par(mfrow = c(1,2))
    plot(residuals(mod_logaritmica2) ~ dimensione[-outliers], data=dataset)
    abline(h=0, col = "red")
    qqPlot(residuals(mod_logaritmica2))

##  6 23 
##  6 20
    par(mfrow = c(1,1))

6.4.4 Unione delle tre tipologie più quella classica, quadratica e Cubica:

    par(mfrow = c(2,2))
    mod_logaritmica <- lm(log(processi) ~ log(dimensione), data = dataset) 
    plot(processi ~ dimensione, data=dataset)
    plot(processi ~ dimensione + I(dimensione ^ 2), data=dataset)
    plot(processi ~ dimensione + I(dimensione ^ 3), data=dataset)

    plot(processi ~ log(dimensione), data=dataset)
    plot(log(processi) ~ dimensione, data=dataset)
    plot(log(processi) ~ log(dimensione), data=dataset)

    par(mfrow = c(1,1))
    summary(mod_logaritmica)
## 
## Call:
## lm(formula = log(processi) ~ log(dimensione), data = dataset)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -2.6441 -0.4204 -0.0970  0.1637  1.8557 
## 
## Coefficients:
##                 Estimate Std. Error t value Pr(>|t|)    
## (Intercept)       6.6319     1.1675   5.680 4.93e-06 ***
## log(dimensione)  -1.3797     0.4603  -2.998  0.00578 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.9188 on 27 degrees of freedom
## Multiple R-squared:  0.2497, Adjusted R-squared:  0.2219 
## F-statistic: 8.986 on 1 and 27 DF,  p-value: 0.00578

7 Calcolo dell’intervallo di Previsione

predict(mod2, newdata = data.frame(dimensione = 10), interval = "prediction") # dove 10 è la dimenisone che vogliamo controllare
##        fit      lwr      upr
## 1 36.30562 21.37121 51.24004

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Formule nel testo: \(\sin(\pi x)\)

Formule su righe separate: \[ \sin(\pi x) \]

Momenti: \[ \mu_k, E(X^k), M_k, m_k, \bar{X}, \bar{x}, \sigma^2, Var(X), S^2 \] Stimatori e stime: \[ \hat \theta, \hat \mu, \hat \sigma^2, \hat \lambda \] \[ \tilde \theta, \tilde \mu, \tilde \sigma^2, \tilde \lambda \] Verosimiglianza: \[ L(\theta), \ell(\theta), \ell'(\theta), \ell''(\theta) \] Frazioni: \[ \frac{1}{\sqrt{2}} X_i, \frac{X_i}{Y_i} \]

Sommatorie: \[ \sum_{i=1}^n X_i, \sum_{i=1}^n \log X_i, \sum_{i=1}^n \sqrt{X_i}, \sum_{i=1}^n \frac{X_i}{Y_i} \] Integrali: \[ \int_{0}^1 f(x) dx, \int_{a}^{\infty} f(x) dx, \int_{-\infty}^{\infty} f(x) dx \]

Come scrivere i comandi R

2 * 3 / log(1.2)
## [1] 32.90889
sin(2 * pi / 20)
## [1] 0.309017
a <- 2 * c(1, 2)
a
## [1] 2 4
z <- qnorm(0.95)
z
## [1] 1.644854
t <- qt(0.95, df = 12)
t
## [1] 1.782288